KINEMATIC SYNTHESIS OF PLANAR, SHAPE-CHANGING RIGID BODY MECHANISMS FOR DESIGN PROFILES WITH SIGNIFICANT DIFFERENCES IN ARC LENGTH.

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KINEMATIC SYNTHESIS OF PLANAR, SHAPE-CHANGING RIGID

BODY MECHANISMS FOR DESIGN PROFILES WITH SIGNIFICANT

DIFFERENCES IN ARC LENGTH

Dissertation

Submitted to

The School of Engineering of the

UNIVERSITY OF DAYTON

In Partial Fulfillment of the Requirements for

The Degree of

Doctor of Philosophy in Mechanical Engineering

By

Shamsul Anuar Shamsudin

Dayton, Ohio

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KINEMATIC SYNTHESIS OF PLANAR, SHAPE-CHANGING RIGID BODY MECHANISMS

FOR DESIGN PROFILES WITH SIGNIFICANT DIFFERENCES IN ARC LENGTH

Name: Shamsudin, Shamsul Anuar

APPROVED BY:

Andrew P. Murray, Ph.D. Advisor Committee Chairman Professor, Dept. of Mechanical and Aerospace Engineering

Vinod K. Jain, Ph.D. Committee Member

Professor, Dept. of Mechanical and Aerospace Engineering

David H. Myszka, Ph.D., P.E. Committee Member

Associate Professor, Dept. of

Mechanical and Aerospace Engineering

James P. Schmiedeler, Ph.D. Committee Member

Associate Professor, Dept. of Aerospace and Mechanical Engineering, University of Notre Dame

John G. Weber, Ph.D. Associate Dean School of Engineering

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c

Copyright by Shamsul Anuar Shamsudin

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ABSTRACT

KINEMATIC SYNTHESIS OF PLANAR, SHAPE-CHANGING RIGID BODY MECHANISMS

FOR DESIGN PROFILES WITH SIGNIFICANT DIFFERENCES IN ARC LENGTH

Name: Shamsudin, Shamsul Anuar University of Dayton

Advisor: Dr. Andrew P. Murray

Design of shape-changing machinery is an area of growing significance. Shape-change may be

employed in the near future to vary the cross section of a wing, create flow-field control by altering

shapes to locally affect downstream fluid behavior, or vary the size of a car seat to meet a wider

array of ergonomic needs. Rigid body shape-change mechanisms offer many advantages including

the high capacity to endure substantial loads while achieving large displacements. Their design

techniques are also well-established. The goal of this research project is to develop the synthesis

theory to address planar rigid-body shape-change where significant differences in arc length define

the problem.

This dissertation presents a process to approximate several design profiles of significantly

differ-ent arc lengths with rigid bodies connected by revolute and prismatic joints. This process is referred

to as segmentation, and the initial step is the conversion of the design profiles into piecewise linear

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This is followed by segmentation which serves to identify the contiguous sets of pieces that

are best approximated by either a rigid bodyM_{-segment or a constant curvature} C_{-segment that} contains a prismatic joint.

To facilitate segmentation, the concept of segment matrix is introduced. A segment matrix

identifies the lengths of the bodies in the sequence of M_{- and}C_{-segments along a profile. The} segmentation process is applied to open, closed, and fixed-end design profiles. A

M

ATLAB-based

tool was developed to facilitate visual assessment of the process and results. Finally, this dissertation

illustrates five mechanization examples that apply the segmentation process, and the fundamental

mechanism synthesis to guide the motion of the chain of rigid bodies to progress to the subsequent

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To my parents, Halimah and Shamsudin,

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ACKNOWLEDGMENTS

Graduate studies at the University of Dayton have been both fun and challenging. The classes

have broadened my perspective as a mechanical engineering graduate, and provided me with the

skills and knowledge needed to become an accomplished engineering instructor, advisor, and

re-searcher. I also found the research group meetings where we shared updates, carried out small

projects, and learned from each other, very beneficial. This interaction has contributed in many

ways to the successful completion of this dissertation which represents the most important

mile-stone on my road to graduation.

I would like to thank Professor Andrew Murray for his guidance, and also his patience during

the extended period that I took to complete this journey. When obstacles came my way, he was

always available with solutions to see me through. Many of my friends are of the opinion that I am

fortunate to have him as my advisor, and I could not agree more.

I am also grateful to the other committee members for their invaluable contribution towards

this research. Professor Vinod Jain guided me in his class and encouraged me to complete my

studies. I thank Professor David Myszka for being supportive in the group meetings and for being

instrumental in the writing of the technical papers we worked on. I also wish to thank Professor

James Schmiedeler for his involvement in my doctoral research work and publications.

I would like to express my appreciation for the ever so generous financial support I received from

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I look forward to serving the country as an educator and sharing the knowledge of machine theory

with the future generations of mechanical engineers.

I am grateful to fellow researchers at the Design of Innovative Machines Laboratory (DIMLab):

Jonathan Lauden, Ali AlMandeel, ‘Chef’ Lin Li, Tong YuXuan, Jo Binjue Li, Dr. Dave A. Perkins,

Andrew Hazlett, Brandon Leedy, Joshua Nieman, Sarotham Reddy and Linda Leben for keeping

me company at the lab and sharing their ideas. I thank Kai Zhao of the University of Notre Dame

for his help in the mechanization issues. I am also indebted to Dr. Aaron Altman and Dr. Markus

P. Rumpfkeil for their insights on aircraft wings. I am grateful to Rise’ Kreitzer, Capt. Edward

Seaman, Constance Norman, Dr. Nicoletta Hary, and all the staff at the Roesch and Marian Library

where I was engaged on a part-time basis.

My family has played a major role in this endeavor and I would like to thank my wife Asmida

and my children Nurul Adlina, Khairul Amri, and Ahmad Syakir for their love, support, and

un-derstanding. I am forever indebted to them all. I am also aware that my parents, Halimah and

Shamsudin, constantly include me in their prayers, and it is my hope that this achievement brings

them much joy.

Lastly, but most importantly, I thank The Almighty God for His countless blessings, and for

pro-viding me with the determination to complete my studies and achieve my goal. With this

achieve-ment I pray for the ability to harness my expertise in this field for the education of future

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II. OPERATIONS ON TARGET PROFILES . . . 26

2.1 Target Profile Generation . . . 26

2.2 Metrics on Target Profiles . . . 31

2.2.1 The Closed-Form Similarity Transformation Solution . . . 36

2.2.2 Example . . . 39

2.3 Curvature Calculations . . . 41

2.3.1 Calculating the Curvature of Target Profiles . . . 41

2.3.2 Curvature Smoothing . . . 42

2.3.3 Regenerating the Target Profile . . . 43

III. THE SEGMENTATION PROCESS . . . 48

3.1 A New Segmentation Process for Target Profiles of Different Arc Lengths . . . . 48

3.1.1 The Segment Matrix . . . 50

3.1.2 M_{-Segments . . . .} ₅₁

3.1.3 C_{-Segments . . . .} ₅₂

3.2 Evaluating and Adjusting the Segment Matrix . . . 54

3.3 Joining the Chain . . . 58

3.4 Compound Segment Types . . . 61

3.5 Other Profile Examples . . . 63

3.5.1 Closed-Profile Example . . . 63

3.5.2 Fixed-End Profile Examples . . . 63

3.6 Comparing the Number of Joints . . . 65

3.7 Versatility of the Segmentation Method . . . 66

3.8 Automating the Segmentation Process . . . 68

IV. SOFTWARE IMPLEMENTATION . . . 72

4.1 Design Profiles . . . 72

4.2 Target Profiles . . . 73

4.3 Segmentation . . . 75

4.4 Visualization of Segmentation Results . . . 77

4.5 Compound Segments . . . 77

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5.1.1 Car Seat . . . 80

5.1.2 U-to-D Transformation . . . 85

5.2 Fixed-End Example . . . 89

5.3 Closed-Loop Example . . . 96

5.3.1 Morphing Wing . . . 96

5.3.2 Flow-Field Control . . . 96

VI. CONCLUSIONS AND FUTURE WORK . . . 106

6.1 Contributions . . . 106

6.2 Future Work . . . 109

BIBLIOGRAPHY . . . 114

APPENDICES: A.

M

ATLAB IMPLEMENTATION OF THE SEGMENTATION PROCESS . . . 123

A-1 Generating Design Profiles . . . 123

A-1.1 Editing Design Profiles . . . 127

A-2 Generating Target Profiles . . . 136

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LIST OF FIGURES

Figure Page

1 An early concept of a compliant mechanism for a morphing wing slat and flap de-signed for the F-111 fighter aircraft [5]. . . 2

2 A morphing wing provides only the amount of lift necessary, reducing the fuel use of the aircraft [3]. . . 3

3 A shape-changing spoiler can be used to improve vehicle stability by providing additional downward force when navigating corners. . . 3

4 The control surfaces in a submarine include (a) the starboard (tail) and (b) aft (front) hydroplanes of (c) the HMS Astute [28]. . . 4

5 An underwater stabilizer uses two SMA actuators to move its flap. (a) The trailing edge is in a neutral position. (b) Joule heating causes the trailing edge to deflect up-ward. (c) Cooling of the top actuator returns the trailing edge to its neutral position [29]. . . 5

6 The shape-changing antenna can alter its shape during the scanning and focusing of a wave [33]. . . 6

7 A soft-top convertible roof [40]. . . 7

8 A mobile stage and roof system in (a) stowed and (b), (c), and (d) are alternate deployed configurations [14, 41]. . . 7

9 A compliant mechanism holding a round object [45]. . . 8

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11 The mechanism is capable of moving the chain from approximating the “U” to the

“D” as it progresses from (a) through (d) based on work in [46]. . . 10

12 The “skin” or outer layer of the shape-changing machine does not have to be rigid. The GINA has the skeleton mechanism underneath made of rigid members [73]. . . 15

13 A cable extends the length of the prosthetic finger and acts as an actuator [83]. . . . 17

14 (a) A mean segment generated for points 1-2-3 of the two different length profiles. (b) Mapping the mean segments on the profiles does not show good results. . . 18

15 A conventional theoretical pressure distributionCpplot of a four-element wing. The leading-edge and trailing-edge flaps are in deployed positions. . . 19

16 The various types of hulls for ships and boats. . . 20

17 Different cross-sectional shapes along a modern ship’s hull [26]. . . 20

18 The Ti-Ni wires return to its original antenna shape by heating [66]. . . 21

19 (a) Schematic diagram of the multiple dies extrusion system. (b) The frame is axi-ally twisted, and the width of the cross section varies steadily [87]. . . 22

20 (a) Schematic diagram of the multiple dies pultrusion system. (b) The cross-sectional shapes vary gradually [88]. . . 22

21 (a) Various cross-section shapes for the two extruded products. (b) The shape-changing die concept [89]. . . 23

22 Profile types include (a) open profiles, (b) closed profiles, and (c) fixed-end profiles. 27 23 Computation time decreases exponentially assdincreases in size. . . 29

24 Design profile (solid) with an approximating target profile (dashed) where points are positioned to give a constant arc length along the design profile. . . 30

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26 (a) Three target profiles, with one deemed the reference profile. (b) Two profiles are transformed to the reference by a similarity transformation (with b = 1). (c) The mean profile. (d) The mean profile transformed back to the original profile locations [46]. . . 33

27 (a) Rigid bodies connected with revolute joints form a chain to closely approximate the profiles. (b) A mechanism design that moves the chain of rigid bodies between the three profiles. . . 34

28 (a) The reference pointsZ1_i. (b) The pointszji to undergo the similarity

transfor-mation. (c) The scaled, rotated and translated z2_i that align with the Z1_i for the optimalb= 0.72,θ =−33.69◦, anddj = [−0.80,0.40]T. (d) The scaled, rotated and translatedz2_i that align withZ1_iforb= 1. . . 40

29 Target profiles (a) with calculated curvature (b) and smoothed curvature (c). The dashed curves in (a) are regenerated profiles and are nearly identical to the original profiles. . . 44

30 The location of˜zji+1is reconstructed from˜zji−1,˜zji,

˜

αji,βj˜i

andrj˜i. . . 46

31 (a) The first segment from three target profiles to be represented by one mean seg-ment. (b) Segments 2 and 3 shifted to reference segment 1 in a distance minimizing transformation. (c) The mean segment is generated as the average of the correspond-ing segment points. (d) The mean segment is shifted back to the original segments in another distance minimizing transformation. . . 52

32 AC_{-segment is built and then transformed to its location in a distance minimizing} configuration. . . 53

33 (a) The second segment from three target profiles approximated with aC_-segment. (b) An arc of the same radius, but varying length, approximates the segment on all profiles. . . 54

34 A rigid-body chain that approximates the target profiles. The chain after the ini-tial segmentation matrix is shown in (a) with Emax = 1.50. The chain after the error-reducing iterations is shown (b) withEmax = 0.38. The inset illustrates that segmentation points are not coincident. . . 57

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36 The joining process unites segmentation points, yet changes the error fromEmax = 0.38toEmax = 0.39. . . 60

37 (a) The original chain consisted of 4 segments resulting inEmax = 0.39, yet one revolute joint exhibited limited motion. (b) The2nd and3rdsegments were fused into a compound segment, resulting in a three-segment chain withEmax= 0.45. . 62

38 (a) The E420 and E850 profiles of airfoils. (b) The chain of ten segments with Emax = 0.015. (c) Fusing segments raises the error toEmax= 0.017while reduc-ing the number of joints. . . 64

39 Fixed-end profiles shown in (a). The final, rigid-body, shape-approximating chain is shown in (b) havingEmax= 0.18. . . 65

40 (a) The [M C M M M_{] chain is selected with an after-assembly error of}_E_max ₌

0.26. (b) A compound segment increases maximum error toEmax = 0.28, which is still lower than the [M C M M_{] design with}_E_max_{= 0}_.₃₇_{in (c). . . .} ₆₇

41 (a) The design profiles. (b) The [C C M C C M M C C C M C C M C_{] segmentation} withEmax = 0.24. . . 69

42 The range from 0 to 0.2 is an acceptable range to be approximated by an arc of constant curvature. . . 70

43 In order to identify a potential region in which to insert another prismatic joint, the curvature distribution (from Fig. 42) is replotted starting with the end of the previously selected curvature band. . . 71

44 (a) A design profile and its corresponding target profile created with 10 points. (b) The plot of the target profile’s arc length as a function of its number of points. . . 74

45 (a) The silhouettes of a small, an average and a large driver [112]. (b) Seat design profiles that ideally suit the small, average and large drivers from (a). . . 81

46 The three seat profiles can be approximated by the five segment shape-approximating chain in (a). The creation of a compound segment reduces the chain to three seg-ments in (b) with only a visually-acceptable increase in error. . . 82

47 (a) The seat accommodates the1st _{percentile group, (b) the}₅₀th _{percentile group,} and (c) the99thpercentile group. . . 83

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49 (a) The same arc length design profiles for U and D. (b) The new profiles include an enlarged “D” so that its overall size is comparable to that of “U.” . . . 86

50 Segmentation shows good approximations with a maximum point-to-point error Emax = 0.27. . . 87

51 (a) The mechanism aligned with the “U.” (b) An interim position close to the “U.” (c) An interim position close to the “D.” (d) The mechanism aligned with the “D.” . 87

52 Labels for the dyads and revolute joints for the UD example. . . 88

53 (a) The NASA 30P30N wing is shown with its slat in the stowed configuration. (b) The slat in the deployed configuration. (c) The slat locations have been used to identify a fixed-end design challenge. . . 90

54 The two rack and pinion systems drive the slat out of the main wing [114]. . . 91

55 (a) Fluctuating instantaneous sound pressure field [115]. (b) Many early types of slats are hinged to the main wing element [116]. . . 92

56 (a) The target profiles generated withsd= 0.1. (b) The four-segment design has an Emax = 0.14. (c) The error minimizing segmentation result with a [C M C] chain andEmax= 0.16. . . 92

57 (a) A fully stowed slat. (b) The slat starts to deploy. (c) An interim position as it approaches the deployed position. (d) The fully deployed (extended) slat. . . 94

58 Labels for the dyads and revolute joints for the wing slat example. . . 95

59 (a) Thin E850 high-speed airfoil and thick E420 high-lift airfoil. (b)-(f) Progression of the morphing wing between E420 and E850 airfoils. The two straight blue lines at the tail are paths for two sliders that guide theC_{-segment. The curved sliders in} red follow a curved path as it guides the other end of theC_{-segment. . . .} ₉₈

60 Labels for the dyads and revolute joints for the morphing airfoil example. . . 98

61 (a) The three profiles of the original flow field example. (b) The new profiles have significant differences in arc lengths. . . 100

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63 (a) The mechanism aligned with the ellipse. (b) An interim position between the ellipse and the tear-drop. (c) The mechanism aligned with the tear-drop. (d) An interim position between the tear-drop and the circle. (e) The mechanism aligned with the circle. . . 103

64 (a) A slider and a slot that can be entrenched in the floor or frame. (b) A telescoping element that can operate from inside the device for flow-field control [118]. . . 104

65 Labels for the dyads and revolute joints for the flow-field example. . . 105

66 (a) The mechanization challenges posed by (a) and (b) are significantly different. Associating the identified point as being at the same location in the segmentation in (c) may be an accurate way of approximating bodies, but poses a significant mechanization challenge. . . 110

67 Initial results for a compliant mechanism with living hinges, approximating three profiles. . . 113

A-1 The first two windows in ShapeChanger. . . 124

A-2 (a) The first design profile is sketched. (b) Identifying that another profile is to be added. (c) The second design profile is sketched. (d) Proceed to the editing of the design profiles. . . 126

A-3 (a) A closed design profile is generated in the workspace with left mouse clicks. (b) A right click closes and finishes the profile. . . 127

A-4 (a) A fixed-end design profile is generated in the workspace with left mouse clicks. The second control point of the next profile is clicked. (b) The second profile adopts the same initial point as the first profile, and the connection to it is automatic. (c) The second profile adopts the same final point as the first profile, and the connection to it is automatic with the right click indicating completion of the profile. . . 128

A-5 Editing operations on the design profiles. . . 129

A-6 (a) A click selects and highlights that profile. A second click indicates that point is to be moved. (b) The curve after a new location is selected for the point. . . 130

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A-8 (a) A click selects and highlights the profile. A second click indicates the range marked by two magenta squares. (b) If the range is right, select the exact location of the new point. Then the new point appears on the profile after the selection. . . . 132

A-9 (a) Select the curve and then the reference point (b) Click on the new location for the selected reference point on the profile to move the entire profile. . . 133

A-10 (a) The top profile is selected, and a second click indicates the control point about which to rotate. A dialog box opens to input the angle of rotation. (b) The curve after a rotation about the selected control point. . . 135

A-11 (a) A pair of profiles is (b) rescaled to have the same arc length. . . 135

A-12 The workspace presents the two target profiles matching well the design profiles beneath them. . . 136

A-13 The relevant information about target profiles influence the next phase, the segmen-tation. . . 137

A-14 A selection of desired piece length,sd = 4, that leads to a poor match between the design and target profiles. . . 138

A-15 Segmentation process takes the input from this window. . . 141

A-16 (a) Reset starting points dialog box. (b) Original starting points of the design pro-files. (c) New starting points for the target propro-files. . . 142

A-17 (a) A user can set whether to see the optimization process changing the segments’ lengths, or (b) to see the result of each chain design, or just see the final result. . . . 143

A-18 A result from the segment synthesis that is selected due to it having the lowest maximum point-to-point error,Emax= 0.09. . . 144

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LIST OF TABLES

Table Page

1 Design vector errors for all five-jointed chains. . . 66

2 Target profile data for the “UDAYTON” example shown in Fig. 41 . . . 68

3 Control points used in constructing the car seat profiles. . . 81

4 The mechanism’s revolute joints for the car seat example in the configuration shown in Fig. 48. . . 84

5 The revolute joints in the segmented chain for the car seat example in the configu-ration shown in Fig. 48. . . 84

6 Control points used in constructing the U and D profiles. . . 85

7 The mechanism’s revolute joints for the UD example in the configuration shown in Fig. 52. . . 88

8 The revolute joints in the segmented chain for the UD example in the configuration shown in Fig. 52. . . 89

9 Control points that define the stowed and deployed profiles of the wing slat. . . 93

10 The revolute joints in the segmented chain for the wing slat example in the config-uration shown in Fig. 58. . . 95

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12 The mechanism’s revolute joints for the morphing wing example in the configura-tion shown in Fig. 60. . . 97

13 Mechanism points for the morphing airfoil example in the configuration shown in Fig. 60. . . 99

14 Revolute joints for the morphing airfoil example in the configuration shown in Fig. 60. 99

15 Points used in constructing the circle, ellipse, and tear drop profiles. . . 101

16 The mechanism’s revolute joints for the flow-field example in the configuration shown in Fig. 65. . . 105

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NOMENCLATURE

M _{Mean segment type}

C _{Constant curvature segment type} cji Length of pieceion design profilej

Cj Arc length of design profilej sji Length of pieceion target profilej

sd Desired piece length for target profiles N Number of points for a segment

n Number of points for a profile in general mj Number of pieces on target profilej nj Number of points on target profilej

mej Number of pieces forethsegment on profilej ne_j Number of points forethsegment on profilej

ǫsj Error representing the difference between the average segment length

and the desired piece length

m∗_j Number of pieces on target profilejthat minimizesǫsj

n∗_j Number of points on target profilejthat minimizesǫsj

Sj Arc length of target profilej

¯

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Z1_i A set ofipoints on a reference profile

zji A set ofipoints on profilejthat will be transformed Z1t A vector composed ofZ1xt=

X

Z1xi andZ1yt=

X

Z1yifori= 1, . . . , N

zjt A vector composed ofzj xt= X

zj xiandzj yt=Xzj yifori= 1, . . . , N b Scale factor in the similarity transformation

θ Angle of rotation in the similarity transformation

Aj Rotation matrix in the similarity transformation

dj Translation vector in the similarity transformation

D The average squared point deviation between a profile or

a segment with its reference

κji Curvature of pointion target profilej

{αji, βji}

T

The center of a circle passing throughzji−1,

zji andzji+1

rji Radius of curvature of pointion target profilej

φ Angle∠zji−1zjizji+1

MAV A micro unmanned aerial vehicle

SMA Shape-memory alloy

SME Shape-memory effect

DOF Degree(s) of freedom for a mechanism

V A design vector that shows the sequence ofM_{- and}C_-segments

SM A segment matrix

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Pji−1 A direction vector that extends fromzji−1 tozji ˜

zji A regenerated profile point based on radius of smoothed curvature

and uniform pieces

˜

rji Radius of smoothed curvature of pointion target profilej ˜

κji Smoothed curvature of pointion target profilej n

˜

αji,βj˜i oT

The circle center for finding˜zji+1

p Number of design profiles

q Number of segments in a chain of rigid bodies α Minimum number of pieces per segment

¯

re Mean radius for theethC_-segments δ Step angle in constructing aC_-segment E_je Error calculated forethsegment on profilej

¯

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CHAPTER I

INTRODUCTION

1.1 State of the Art in Shape-Change

Machines can benefit from the capacity to vary between specific shapes in a controlled manner.

Although there are numerous examples of conventional machines that exhibit shape-change, novel

approaches are currently being developed to solve increasingly sophisticated problems and address

a variety of new applications. This chapter presents an overview of this research work and places

the new research generated in this dissertation in its context.

1.1.1 Airfoils and Blades

The focus of much study has been on shape morphing aircraft wings that increase performance

over a range of flight conditions [1, 2]. Most of the design work has centered on changes between

wing profiles of similar arc length [3, 4, 5]. The fundamentals of aerodynamics dictate, however,

that lift and drag can be significantly altered with a change in camber and chord [6]. That is, for

high lift situations (e.g., approach, landing, and climb), a higher camber and longer absolute chord

are desirable, whereas for efficient cruising, a lower camber and shorter chord are desirable. Kota

et al. [5] discovered an early shape-changing mechanism design using rigid-links and fiberglass

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superior aerodynamic benefits, as shown in Fig. 1. However, in that work, no significant change in

arc length of the airfoil was considered.

Figure 1: An early concept of a compliant mechanism for a morphing wing slat and flap designed for the F-111 fighter aircraft [5].

Abdulrahim et al. [7] studied a membrane wing for a micro unmanned aerial vehicle (MAV).

The under cambered membrane wing has a wingspan of 24 inches. Such a wing changes its shape by

morphing its entire profile by wire control rods connected to small motors in the fuselage. Limaye et

al. [8] incorporated compliant kits of beams and connectors to model the trailing edge of a morphing

aircraft wing. The compliant beams are deformed to alter the shape of the wing. Strelec and

Lagoudas [9] used smart materials to control the shape of a reconfigurable wing to match a set

of desired airfoil profiles. Weisshaar et al. [10, 11] underscored the importance of searching for new

technologies to gain improved control surfaces for military aircrafts. They used smart materials to

actuate the shape change in the trailing-edge flap of an aircraft wing. Kota et al. [5, 12] and Santer

et al. [13] used compliant mechanisms to change the shape of a control surface that approximates

an airfoil. Kota et al. [12] reported an increase in lift while maintaining a nearly constant drag

coefficient. Another shape-changing wing design is illustrated in Fig. 2 in which the wing morphs

shape as the aircraft burns fuel and less lift is required [3]. In addition to aircraft, airfoils may be

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of rigid-body segments. The spoiler can actively change shape as the need for traction varies. On

straight stretches of highways, lower traction caused by a low pitch angle in the spoiler results in

better fuel efficiency, whereas in curves, a high pitch angle in the spoiler produces higher downward

force for better grip on the road [15].

Figure 2: A morphing wing provides only the amount of lift necessary, reducing the fuel use of the aircraft [3].

Figure 3: A shape-changing spoiler can be used to improve vehicle stability by providing additional downward force when navigating corners.

Cesnik et al. [16], Wong [17], and Grohmann et al. [18] investigated the use of smart materials

like active fiber composites and piezoceramics to morph the shapes of helicopter blades. The US

military, Sikorsky Helicopters, and DARPA are among organizations interested in developing

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and trailing edges of the airfoil to change blade shapes according to the speed of the wind. The

anticipated optimized performance achieved by integrating active flow control in the wind turbine

rotors is presented in [23, 24].

Hydroplanes (sometimes called “underwater wings”) and rudders can benefit from shape-changing

[25]. In surface ships, active fin stabilizers or gyrofins, play a major role in preventing capsizing

due to pitching and rolling moments [26]. Molland [27] explained that as a submarine operates

underwater at a speed above ten knots, its stability and control depend predominantly on the

hy-drodynamic forces and moments on the hull and control surfaces. To remedy these problems, two

sets of hydroplanes are needed. Figure 4 illustrates their locations. Gillmer mentions that some

un-(a) (b) (c)

Figure 4: The control surfaces in a submarine include (a) the starboard (tail) and (b) aft (front) hydroplanes of (c) the HMS Astute [28].

derwater stabilizers can have trailing-edge flaps that produce the most effective vertical force [26].

To control such flaps on underwater stabilizers, Garner et al. [29] presented the development of a

biomimetic active hydrofoil that utilized shape-memory alloy (SMA) actuator technology. They

investigated the use of 0.58 mm diameter wires made of nickel-titanium with 10% copper on the

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(a) (b)

(c)

Figure 5: An underwater stabilizer uses two SMA actuators to move its flap. (a) The trailing edge is in a neutral position. (b) Joule heating causes the trailing edge to deflect upward. (c) Cooling of the top actuator returns the trailing edge to its neutral position [29].

1.1.2 Ship and Boat Hulls

Gernez et al. [30] developedProtei, which is an autonomous, shape-shifting, sailing vessel for

ocean exploration and conservation. The flexible and segmented vessel resembles an airfoil when

viewed from the top. The large rudder underneath is merged with the hull and changes shape to

curl right or left as the boat captures the forces from the wind and sea currents. In these marine

applications, different arc length shape-change may be practical to achieve the desired shapes.

1.1.3 Antenna and Mirror

Advances in electro-optics resulting from shape change include active aperture antennas [31]

and deformable mirrors [32]. A compliant shape-changing antenna designed by Lu et al. [33, 34] is

shown in Fig. 6. Furthermore, Plecnik [35] proposed a chain of rigid-bodies with reflective surfaces

that can be adjusted to increase or decrease light intensity. Gupta [36] studied the application of

piezoelectric actuators to steer a satellite antenna. The study used a Genetic Algorithm to optimally

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Figure 6: The shape-changing antenna can alter its shape during the scanning and focusing of a wave [33].

Satellites rely on the shapes of mirrors to capture clear images of the universe. Maji et al. [37]

highlighted methods by which mirror pieces can be situated in the right position and orientation

using smart material alloys that can be actuated to change shape via electrical signals. Stamper et

al. [38] used membrane mirrors like electroformed nickel that are tensioned at discrete attachment

points. Fern´andez and Artal [39] used a membrane deformable mirror with 37 electrodes to study

its use as an adaptive optics element.

1.1.4 Convertible Structures

Automotive convertible roofs [40] and portable performance stages [14] involve shape changes

in structural applications. Figure 7 shows a side view of a fully deployed convertible roof of a car.

A shape-change problem may be defined by the deployed and stowed configurations of such a roof.

Erdman et al. [14] included the analysis problem of a portable performance stage that is shown

in Fig. 8. The stage was designed to be large enough for live performances with an integrated

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[image:30.612.194.455.304.627.2]

Figure 7: A soft-top convertible roof [40].

(a) (b)

(c) (d)

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1.1.5 Robotic Gripper and Prosthetics

In manufacturing, robotic grippers are designed to move between prescribed shapes. Mohd

Zu-bir et al. [42] used a compliant mechanism for a microgripper. Hirose and Umetani [43] developed

an active shape-conforming gripper that could wrap around objects with round and concave shapes.

A variable arc length gripper would be desirable to grasp objects of different sizes. Moon [44]

used compliant mechanisms to approximate the shape of a human finger during a gripping motion.

Petkovic et al. [45] used carbon-black filled silicon material that is able to morph the gripper to

accommodate different objects, as shown in Fig. 9.

Figure 9: A compliant mechanism holding a round object [45].

1.1.6 Artwork

For entertainment purposes, a mechanism that morphs between the letters “U” and “D” has been

proposed [46]. The profiles and the set of rigid bodies that morph between the two letters are shown

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(a) Profiles (b) Segments

Figure 10: (a) Design profiles define the shapes of the letters. (b) A chain of rigid-bodies connected by R joints approximates the two letters.

1.2 Various Shape-Changing Methods

There have been many emerging technologies developed for various applications. The main

ones include rigid-body mechanisms, compliant mechanisms, shape-memory effect materials and

membranes, as well as combinations of these with rigid-bodies.

1.2.1 Rigid-Body Shape-Change

Rigid body mechanisms are typically composed of familiar machine elements like linkages,

gears, cams, axles, etc. The theories governing their analysis are well established, explaining

accu-rately their kinematics and the forces encountered [47]. The design of systems composed of these

elements also has many well established methodologies, with many of the components

[image:32.612.180.468.79.310.2]

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(a) (b)

[image:33.612.168.478.98.616.2]

(c) (d)

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and design theories, rigid-body mechanisms readily produce large displacements while delivering

high load-carrying capacities. Potential drawbacks include backlash, wear, increase in part-count,

weight, assembly cost and time, and routine maintenance [48]. As rigid-body mechanisms are

com-mon and encounter significant use, these drawbacks are well addressed in most machine designs

[49, 50].

In this work, the rigid-body shape-changing mechanisms are composed of bodies connected by

revolute (R) and prismatic (P) joints. Examples of rigid-body shape-changing designs from Section

1.1 include the morphing wing in Fig. 2 and morphing car spoiler in Fig. 3, the convertible car

roof in Fig. 7, the mobile performance stage in Fig. 8, and the artwork example in Fig. 11. As the

complexity of some of these devices may indicate, an additional design challenge in several of these

examples is the identification of layering of the links in order to avoid interference [51, 52].

Generally, rigid-body mechanisms can be driven with a single actuator [14]. The input force or

torque can be applied to one of the links of the mechanism [53]. Actuation of rigid-body

mecha-nisms is typically achieved with various kinds of electric motors, ball screws, hydraulic and

pneu-matic cylinders, springs and elastic bands, and occasionally, human-power [14]. Electric motors like

AC motors, DC motors, geared motors, stepper motors, and servomotors provide rotary motion that

can be attached to a driving axle. Some motors are integrated with ball screw mechanisms to provide

linear motion. Hydraulic and pneumatic systems use hydraulic oil and air respectively to initiate

lin-ear motion in cylinders. Manual actuation by cranking and levering also see common use in many

mechanisms. Nonetheless, output motions can also be achieved by having multiple actuators such

as in many industrial robots. Kota and Erdman [54] argued that this is usually not an efficient use

of energy and should be considered primarily if flexibility in motion is required. In addition to the

widely-used actuators mentioned previously, mechanisms can be actuated by shape-memory alloys

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Due to their well established principles and components, rigid-body mechanisms offer a

solu-tion worthy of considerasolu-tion in solving shape-change problems. Rigid-body shape-changing

mech-anisms have the potential to benefit from qualities such as high stiffness and large load carrying and

displacement capabilities.

1.2.2 Compliant Mechanisms

Shape-change may be accomplished by using compliant mechanisms, which can be designed in

a manner similar to rigid-body mechanisms [58]. Without hinges, though, compliant mechanisms

have the advantage of providing a smooth morphing boundary without discontinuities. Compliant

mechanisms usually have qualities of both a mechanism and a structure [34]. Howell introduced the

concept of the pseudo-rigid-body model to aid the design of compliant mechanisms [47]. Another

method used in their synthesis is topology optimization, which attempts to solve a design problem

by iteratively removing material from regions in a structure that are less sensitive to reducing the

overall structural stiffness and thus change its final shape [33, 34].

Section 1.1 lists many compliant shape-changing devices including a prosthetic hand, morphing

aircraft wing in Fig.1, and robotic gripper as in Fig. 9. Shuib et al. [48] show other applications

including compliant staplers and knee-joints. Additionally, there are also many biologically inspired

technologies including those for flapping wings [59] and artificial biomimetic robotic swimming fish

[60].

Goldfarb and Speich [61] mention that a benefit of compliant mechanisms is the absence of

revolute joints and bearings, thus eliminating mechanical backlash (from clearance) and Coulomb

friction (which causes wear). Parlaktas [62] added other potential advantages of compliant

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and less noise. Mankame and Ananthasuresh [63] note that compliant mechanisms have the

advan-tages of being a single-piece and utilizing elastic deformation of the mechanism bodies instead of

kinematic joints, which is very useful at the micro level since they do not require assembly.

Com-pliant mechanisms also store elastic energy as they deform, so there is no need for springs to return

the mechanism to its original position. Consequently, this also means they require a holding force

in a deformed position.

Other disadvantages of compliant mechanisms include the relatively small range of motion,

imprecise axis-drift motion, off-axis stiffness, and high stress-concentration at the joints [61, 64].

Converting reciprocating translation to rotational motion is also a problem that requires alternative

approaches such as a contact-aided compliant mechanism [63]. The displacement achievable with

a compliant mechanism is limited by the elastic properties of its composing material, so it is quite

difficult to accommodate shape changes involving significant differences in arc length.

1.2.3 Membranes and Smart Materials

In space, a very large, deployable telescope can be made of segments of stretched reflective

membranes [37, 38]. Being lightweight and flexible, they can be transported in a compact

configura-tion. Section 1.1.3 included a coated deformable membrane mirror that changes shape significantly

when deployed [17, 32, 39]. Typically, after a mirror is deployed, the flatness of its flexible surfaces

requires correction. This may be accomplished via the use of electrostatic, electromagnetic, electron

gun or boundary actuation techniques in a closed-loop system [38, 65].

In another space application, Otsuka et al. [66] showcased a nickel-titanium alloy for a space

antenna as shown in Fig. 18. Smart materials typically change shape via an applied voltage, when

heated, or due to changes in magnetic force, humidity, or pH value [67, 68, 69, 70]. They can

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is the shape-memory alloy (SMA), which may be classified as “way” or “two-way”. The

one-way materials may be deformed to change shape at low temperature. When heated above a certain

temperature, they start to revert to their original shape in the austenitic phase. A two-way SMA

material can assume two pre-trained shapes by certain thermomechanical treatments that depend on

heating or cooling at certain temperatures. This represents a limitation should there be more shapes

to be approximated. Research in wings and helicopter blades that utilize smart materials typically

incorporate piezoceramics or active fiber composites such as in Weisshaar et al. [10, 11], Cesnik et

al. [16], Wong [17], and Grohmann et al. [18].

1.2.4 Combinations Involving Membranes and Rigid Bodies

Many systems considered for shape change require smooth design profiles. However, rigid-body

rotation about discrete revolute joints will result in a profile with abrupt continuity changes that

may hinder the approximation of a smooth curve. Recent advancements in membrane technology

for mirrors, aeronautics, and automotives could be coupled with rigid-body mechanisms to address

this issue [72, 57]. For instance, BMW is investigating shape-changing cars that are covered with

plastic-coated lycra fabric that is stretched over a metal frame with moving components. This

concept car, GINA [73], is shown in Fig. 12.

1.3 Rigid-Body, Shape-Changing, Mechanism Design Methodology

This section overviews the established design process for planar rigid-body shape-changing

mechanisms that includes design profile specification, target profile generation, segmentation, and

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[image:38.612.247.403.63.214.2]

Figure 12: The “skin” or outer layer of the shape-changing machine does not have to be rigid. The GINA has the skeleton mechanism underneath made of rigid members [73].

1.3.1 Profiles and Segmentation

Planar rigid-body shape-changing mechanism design begins with a set of design profiles, such as

airfoil profiles for loiter and attack modes. The synthesis process proceeds by representing each of

the design profiles in a standardized manner, such that comparisons can be made among them. This

standardized representation is a coordinated set of points on the design profiles defining a

piecewise-linear curve that is termed the target profile. The design process continues with a segmentation phase

that creates segments, which are generated in shape and length so that they form a jointed chain

of rigid bodies that approximate corresponding segments on each target profile. To complete the

synthesis, a mechanization phase typically adds binary links to each segment in order to achieve the

lowest degree-of-freedom (DOF) linkage possible. Although this established process of designing

a rigid-body shape-changing mechanism remains true for this work, substantial changes are needed

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1.3.2 Mechanization Methods

A system with a low DOF is commonly preferred for simplicity in control [74, 75]. The

mechanization phase involves adding rigid constraining links and joints, forming a mechanism that

smoothly transitions the shape approximating chain between the target profiles with a limited

num-ber of actuators. In many applications, the reduced cost and control requirements of fewer actuators

outweighs the kinematic complexity. When a single-DOF system is desired and the number of

tar-get profiles is less than or equal to five, it is theoretically possible to add binary links without further

increasing the profile matching error. The dimensional synthesis task for rigid body guidance

iden-tifies appropriate circle points on the rigid links of the shape approximating chain and center points

on the frame. Machine theory texts, such as McCarthy [76], provide methods for dimensional

synthesis for rigid body guidance. However, it was revealed through experience that eliminating

circuit, branch and order defects becomes problematic when more than three profiles are involved.

Balli and Chand [77] provided a thorough discussion on solution rectification. Consequently, for

two and three profiles, the mechanization of shape-changing linkages has been accomplished by

adapting dimensional synthesis techniques to Geometric Constraint Programming (GCP) with a

computer-aided design package as in Kinzel et al. [78].

For greater than three profiles, mechanization of rigid-body shape-changers is performed as

outlined in Murray et al. [46, 79]. Least-squares approximations such as those developed by Yao

and Angeles [80] can be used to locate circle and center points for each segment. The structural

error associated with such approximate motion synthesis methods will further increase the shape

approximating error. A search algorithm is implemented to examine many circle and center point

pairs, designating candidate designs as those that produce an acceptable level of structural error.

The candidate designs are then evaluated to determine whether they can be actuated monotonically

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can be ranked by a quality factor of the designer’s choosing. This search approach does not yield

optimal designs in any formal sense, but produces a number of viable designs that can be evaluated

according to various metrics. Expanding on the search process, Zhao et al. [81] illustrate how

genetic algorithms can be used to synthesize planar rigid-body shape-changing mechanisms. Once

a successful mechanism has been formed, it may benefit from the addition of a coupler driver to

reduce actuator effort and avoid mechanism defects [82].

The mobility of a final design is not necessarily restricted to a single DOF or being driven by

linkages formed from rigid bodies. There are many other possibilities of actuating a chain of

rigid-bodies from one position to another. Cable and spring mechanisms [83, 84], smart-material

actu-ators like piezoceramics [71], and multiple-motor actuation [54] can all be considered. Figure 13

shows a segmented prosthetic finger that can bend into a curled shape by tensioning its cable [83].

Alternately, robotic systems are typically serial links actuated at every joint. This dissertation does

not focus on the mechanization stage of the work and an ad-hoc method is used to complete

sev-eral examples in Chapter V. Regardless of the fact that a mechanization is shown, the chain may

be controlled in a variety of ways with additional consideration given to the appropriate degrees of

freedom.

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1.3.3 Limitations of This Method

When two profiles are not similar in arc length, the resulting mean profile does not approximate

either of the two profiles well. This, however, depends on the significance of the difference in arc

lengths. Figure 14a illustrates the mean segment generated for points 1 through 3 of the profiles,

whereas Fig. 14b shows two copies of the mean segments being placed on top of the two original

segments. Obviously, the mean segments do not approximate any portion accurately. Another

problem is that the method is not capable of assigning a segment type that can change length.

Hence, there is a need for another method to solve different arc-length problems.

(a)

(b)

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1.4 Shape-Changing for Different Arc-Length Profiles

Some examples of shape-changing mechanisms that have different arc-length profiles include

wings, boat hulls, satellite antenna, and forming dies. This dissertation proposes new methodologies

for the design of rigid-body shape-changing mechanisms that approximate design profiles with

sig-nificant differences in arc length. This affects both the profile generation and segmentation phases.

1.4.1 Different Arc Length Examples

Many large aircraft wings have multiple elements to achieve the required lift and drag for

dif-ferent flight modes. When in operation, some of these elements are deployed to effectively extend

the profile of the wings. Figure 15 shows a pressure distribution plot for a four-element wing [85],

which is an example of different arc length profile for a shape-changing mechanism.

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Wong [17] highlights the idea of a shape-changing boat hull that can affect maximum speed,

ride comfort, and fuel consumption. Several basic foil shapes used by naval designers are shown

in Fig. 16. These also show the need for different arc-length shape-changing mechanisms since

the profiles’ perimeter are significantly dissimilar. Gillmer [26] explained that modern ship designs

employ modifications of these throughout sections of the hull as illustrated in Fig. 17. While overly

ambitious for large sea-going ships, a shape-changing hull is an attractive concept that would enable

much smaller boats to make adjustments for the navigation of seas, rivers, lakes, swamps etc.

[image:43.612.255.391.443.572.2]

(a) Planing (b) Deep hull (c) Multi hull (d) Round hull (e) Deep vee

Figure 16: The various types of hulls for ships and boats.

Figure 17: Different cross-sectional shapes along a modern ship’s hull [26].

Goodyear Aerospace Corporation studied the use of Ti-Ni shape-memory effect (SME) wires

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spatial environment. Figure 18 shows the progression as the deformed wires revert back to the

original shape of the antenna through solar heating.

(a) Deformed shape (b) Interim shape 1

(c) Interim shape 2 (d) Fully deployed shape

Figure 18: The Ti-Ni wires return to its original antenna shape by heating [66].

Altering an extruded plastic profile by sliding segments of the die orifice in a controlled manner

[86] may also be considered a shape change. Choi et al. [87] investigated the forming of aluminum

frames for automotive applications through the use of a CNC variable section extruder. Figure 19

shows the schematic diagram of their experimental setup and the finished product that has different

cross-sections at opposite ends. Note that the perimeters of the sections exhibit drastically different

arc lengths. O’Connor proposes a method to produce a pultruded thermoplastic composite body

having at least two integral sections of different cross-sectional shapes [88]. The system arranges

different sets of dies in series as illustrated in Fig. 20a. Potential varying cross-section pultrusions

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(a) (b)

Figure 19: (a) Schematic diagram of the multiple dies extrusion system. (b) The frame is axially twisted, and the width of the cross section varies steadily [87].

(a)

(b)

[image:45.612.207.445.327.627.2]

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Murray et al. studied the design of a shape-changing extrusion or pultrusion die [89]. Different

arc-length profiles can be seen in Fig. 21a and the extrusion machine that is designed to produce

such shapes is shown in Fig. 21b.

(a) (b)

Figure 21: (a) Various cross-section shapes for the two extruded products. (b) The shape-changing die concept [89].

1.4.2 New Profile Generation

The curves that are specified as the desired shapes to be matched by the edge geometry of rigid

links are called design profiles. Points are generated along each design profile that when connected

by line segments, approximate the design profiles. These new curves are called the target profiles.

The line segment connecting two adjacent points is called a piece. Correspondingly, the number

of pieces will be one less than the number of points on each target profile. Since profiles can have

different arc lengths, the number of points used may not be identical from one design profile to

another. Murray et al. [46] assign the same number of points to each target profile. With significant

differences in arc length, this work assigns a different number of points to each profile, distributed

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allow for the inclusion of rigid bodies connected with prismatic joints, which can vary the length.

As the desired piece length is made smaller, the generated profiles better approximate the shapes of

the design profiles.

1.4.3 New Segmentation Method

Segmentation is the second phase of the design process. The goal of this phase is to create

segments that are generated in shape and length so as to form rigid links that approximate

corre-sponding segments on each target profile. Connecting these segments with revolute joints yields

a chain of bodies that approximates the target profiles and hence, the design profiles. There are

two basic types of segments, the mean segment (M_{-segment) and the constant curvature segment} (C_-segment).

When a segment contains the same number of points on each target profile, an M_-segment is generated. The segments are transformed to the corresponding segment on a reference profile

in distance minimizing configurations. The M_{-segment is generated as the geometric center of} each set of corresponding points before being transformed back to approximate the corresponding

segments. A poor match may be improved by reducing the number of points being matched or

changing start and end points.

When a different number of points on each target profile are to be approximated, aC_-segment is generated. By calculating an average radius of constant curvature and an average piece length, a

segment of constant curvature is generated. As with theM_{-segment, a poor match may be improved} by reducing the number of points being matched or changing start and end points.

M_{- and}C_{-segments that approximate all pieces on all target profiles are assembled by joining} their end points with revolute joints. These assembled chains are repositioned to best match each

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-andC_{-segments may be changed, the length of each segment may be changed, or the number of} segments may be increased.

In addition to the two basic types of segments, a third type of segment may be formed by fusing

anM_{- and a}C_{-segment or two}C_{-segments. Segments are fused after they are joined as a chain} and aligned with the profiles. Should the relative rotation about an R joint be small, fusing of the

segments is considered. Note thatM_{-segments are not fused with other}M_-segments.

1.5 Organization

With the goal of segmentation of a set of design profiles of significantly different arc length

into a chain of rigid-bodies connected by R and P joints, this dissertation is organized as follows.

Chapter II discusses the creation of target profiles when the arc lengths of the design profiles are

significantly different. Having created the target profiles, Chapter III explains the generation of

M_{- and}C_{-segments. The}

M

_{ATLAB software implementation that enables the design of segments} to be carried out is explained in Chapter IV. Chapter V discusses several mechanization examples

in which different profiles serve to show the utility of the methods developed in this work. Lastly,

conclusions and suggestions for future work are furnished in Chapter VI. A user’s guide in Appendix

A provides examples of how to use the software to create a chain of jointed segments from a set of

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CHAPTER II

OPERATIONS ON TARGET PROFILES

The first goal of this chapter is to showcase the process for creating target profiles from a set

of design profiles. Design profiles specify the design challenge in that they are the set of curves

used to describe the shapes that the shape-changing mechanism should approximate. Target profiles

are the set of curves generated from the design profiles but defined by points spaced to facilitate

approximation by a connected chain of rigid links. Once generated, target profiles replace the

design profiles and are used in all following processes to create the chain of links. The second goal

of this chapter is to establish the metrics by which distance between target profiles or their segments

can be determined.

2.1 Target Profile Generation

The challenge posed by shape-change is the specification of a set ofpdesign profiles that rep-resent the different shapes to be attained by the mechanism. Murray et al. [46] define a design

profilejas an ordered set ofnjpoints for which the arc length between any two can be determined. Figure 22 exhibits the three types of design profiles considered: open, closed, and fixed-end

pro-files. While the illustrative examples shown in this work are for the open profile case, this research

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(a) (b)

[image:50.612.151.503.81.350.2]

(c)

Figure 22: Profile types include (a) open profiles, (b) closed profiles, and (c) fixed-end profiles.

Given the definition of design profiles, they may be viewed as being piecewise linear. A piece

is the line segment connecting two contiguous points on a profile. Theithpoint on thejthdesign profile is designated{aji, bji}

T

. The length of theithpiece on thejthdesign profile is

cji = q

aji+1−aji 2

+ bji+1−bji 2

, (1)

and the arc length of the entirejthdesign profile is

Cj = nj−1

X

i=1

cji. (2)

The design profiles may be defined by any number of points spaced at various intervals, producing

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As previously defined, target profiles are a set of curves that represent the set of design profiles.

They have common features so that groups of contiguous points can be compared to all profiles in

order to form a suitable chain of rigid bodies that when repositioned will approximate all design

profiles. In earlier work [46, 79], the design profiles were assumed to be of roughly equal arc

lengths, C1 ≈ C2 ≈ . . . ≈ Cp, and each target profile can be formed by distributing the same n number of defining points equally along the corresponding design profile. The target profile

becomes a piecewise linear curve composed of pieces with roughly the same length, cji ≈ ckl,

∀i, j, k, l. Constant piece lengths allow for identification of corresponding points on each target profile.

The general profiles discussed in this work may possess substantial differences in arc length.

Using the same number of points on different length profiles would result in different piece lengths

and contaminate the shape comparisons among groupings of contiguous points. In order to produce

a constant piece length, the conversion scheme from design to target profiles must be modified from

the established method to allow for a different number of points on each target profile. By specifying

a desired piece lengthsd, the number of piecesmj on profilejcan be determined. Smaller values ofsd will produce more pieces and typically result in smaller variations between the design and target profiles. Experiments have shown as the number of pieces increases, the time it takes to

complete the synthesis increases exponentially. Hence, sd needs to be controlled to achieve the desired accuracy in an acceptable computation time. Figure 23 shows the exponential relationship

betweensdand the computation time done on a machine with IntelR CoreTM i5 M540 processor at 2.53GHz with 4.00 GB total memory. The usable space, however, is 2.98 GB and the system is

32-bit. This experiment used the three-profile example with four segments that is shown in Fig. 25.

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[image:52.612.176.470.66.308.2]

Figure 23: Computation time decreases exponentially assdincreases in size.

The number of pieces must be an integer, and an initial value is calculated as

mj =

Cj sd

, (3)

where⌈ζ⌉ represents the ceiling function, the smallest integer not less thanζ. Provisional target profiles are generated by distributingnjpoints at increments ofCj/mj along thejthdesign profile. A distribution of target profile points along a design profile is shown in Fig. 24. The jth target profile becomes a piecewise linear curve connecting the ordered set of pointsz_j

i = {xji, yji}

T ,

i= 1, ..., nj. The length of theithlinear piece on thejthtarget profile is

sji =

zji+1−zji

=

q

xji+1−xji 2

+ yji+1−yji 2

. (4)

The corresponding number of points on the target profilejisnj =mj+ 1.

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j=1

)

,

(

1 1 1 1

y

x

)

,

(

2 2 1 1

y

x

)

,

(

3 3 1 1

y

x

)

,

(

5 5 1 1

y

x

)

,

(

4 4 1 1

y

x

2 1

s

C

1

/m

1

1 1

s

3 1

s

4 1

s

Figure 24: Design profile (solid) with an approximating target profile (dashed) where points are positioned to give a constant arc length along the design profile.

of the design profile results in piece lengths shorter thansd, as seen in Fig. 24. That is, some piece sji = sdonly whensd divides exactly intoCj andthe design profile has a zero curvature portion

long enough to include the entire piece. The average piece length for thejthprofile is

¯

sj =

1

mj





nj−1 X

i=1 sji



. (5)

As the provisional target profile is constructed to (potentially) have too many pieces to

accu-rately achievesd, the likely scenario is that fewer pieces will produce a value of ¯sj closer to sd. An error representing the difference between the average segment length and desired piece length is

calculated asǫsj =|sd−s¯j|. Decreasingnj by 1 and redistributing points along the design profile

creates a new target profile. Points are removed untilnj = n∗_j (and correspondingly,mj = m∗_j) that minimizesǫsj. The end result is the fewestn

∗

j points are used to construct thejthtarget profile such that all linear piece lengths are approximately equal to the desired piece length. Desirable

target profiles are those with the fewest pieces that achieve the accuracy needed to satisfactorily

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approximation of the design profiles. Conversely, as the calculations presented in the later sections

are dependent on this number, having fewer pieces reduces computation time.

After eachm∗_j is established, the total length of thejthtarget profile is calculated as

Sj = m∗

j X

i=1

sji. (6)

Applying this process to all design profiles, p target profiles are constructed such that all linear pieces have lengths that are approximately equal tosd. The average length of allm∗j linear pieces on allpprofiles is

¯

sm=

Pp

j=1 Sj

Pp

j=1m∗j

. (7)

If the representation of the design profiles lacks the desired accuracy, a smaller desired piece length

may be used to increase the number of points defining the target profiles.

Three target profiles are shown in Fig. 25. The lengths of the design profiles areC1 = 26.1738, C2 = 31.0847, andC3 = 34.4737. For a desired piece length ofsd = 0.35, the target profiles

have m∗1 = 76, m∗2 = 90, and m∗3 = 99 pieces. The average piece length issm¯ = 0.3499and the lengths of the target profiles areS1 = 26.1669,S2 = 31.0760, andS3 = 34.4562. Although S1/C1 = 0.9997,S2/C2 = 0.9997, andS3/C3= 0.9995in this example, the heuristic in this work

is to setsdsuch thatSi/Ci ≥ 0.99for all profiles in the set unless the design problem dictates a specific accuracy. Values are reported in this initial example to four decimal places to highlight the

minor differences between design and target profile lengths and desired and achieved piece lengths.

The four decimal places are not meant to suggest that these are significant figures.

2.2 Metrics on Target Profiles

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0 5 10 15 20 25 30 0

2 4 6 8 10 12 14 16 18 20

j = 1

[image:55.612.143.505.74.328.2]

j = 2 j = 3

Figure 25: Three design profiles with significantly different arc lengths represented with target profiles of nearly constant piece length.

these profiles is determined. Three such profiles are shown in Fig. 26a. Denoting the points on

the “reference profile” asZ₁

i and those on one of the other two profiles (the dash-dot profile, for

example) as z_j

i, the values of rotation matrix Aj and displacement vectordj are calculated that

solve Eqs. (8) and (9) under the restriction of scale b = 1. This restriction is necessary due to the consideration of these points as representing rigid bodies. Figure 26b shows both of the target

profiles aligned with the reference profile in this way. Continuing with the design of a rigid-body

shape-changing mechanism, a “mean profile” is generated from the geometric center of each Z₁

i

and the shiftedz_j

i, as shown in Fig. 26c. Using additional similarity transformations, the mean

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all of the target profiles in order to create a mean segment and to move this mean segment back to

the locations nearest the original target profiles.

(a) (b)

(c) (d)

Figure 26: (a) Three target profiles, with one deemed the reference profile. (b) Two profiles are transformed to the reference by a similarity transformation (withb= 1). (c) The mean profile. (d) The mean profile transformed back to the original profile locations [46].

If the design goal were to find a single body to approximate the three target profiles, the

[image:56.612.199.450.148.523.2]

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in Fig. 26d can be determined. Should this fail to reproduce the original profiles with adequate

accuracy, the process described above can be performed on individual portions of the target profiles.

Figure 27a shows an example in which the three target profiles are considered to consist of four

seg-ments. Each segment now does a better job of approximating a shorter section of the original target

profile, and combining these segments into a single chain connected by revolute joints produces a

more accurate matching of the original target profiles. Figure 27b shows the complete mechanism

that can be used to move the chain between the three profiles. Note that increasing the number of

segments increases the complexity (the number of links) of the mechanism that guides the segments

along the three profiles. The details of this process are found in Murray et al. [46] and Persinger et

al. [79].

(a) Segmentation (b) Mechanization

Figure 27: (a) Rigid bodies connected with revolute joints form a chain to closely approximate the profiles. (b) A mechanism design that moves the chain of rigid bodies between the three profiles.

The application of a similarity transformation in mechanism analysis is not far different from

the needs in the image registration community. Zitov´a and Flusser [90] identified a critical step in

image registration defined as “transform model estimation.” In this step, one attempts to identify the

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reference image. In planar cases, this mapping function is restricted to be a similarity transformation

that can be solved in closed-form as

Zji =bAjzji+dj, (8)

wherezji are the points in the sensed image,bis a scaling factor,Aj is a rotation matrix, anddj is

a displacement vector. The desired values ofb,Aj, anddj are those that minimize

D(Aj,dj, b) =

1

N N